Abstract
Nature is believed to be organized by a mathematical fundamental structure. Therefore, the experiments are interpreted through mathematical models. Unfortunately, experiments can only provide macroscopic outputs, even when referred to quantum elementary object. Starting from such observation, I first consider the concept of anomaly as groundbreaking information to falsify a theory. The separability of a system between an experimental equipment and a microscopic object is discussed. Non commutative microscopic observables of elementary entities are postulated from a set of measurements of macroscopic observables interpreted as their eigenvalues. I explain the major role of the Gelfand-Naimark-Segal construction of the representation of classical and quantum abstract $C^*$-algebras to recognize the impossibility of building a theory with a unified domain for the microscopic and the unavoidable macroscopic observables. I discuss implications of the Gelfand theorems on both macrorealism emergence from coarse grained measurements and decoherence programs. Finally I apply the results to determine the fundamental impossibility to identify a Theory of Everything with the mathematical structure attributed to Nature.
Enrico Prati